Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution was kind of different from the solution provided by the following website. The solution in the website is much more beautiful and more concise than mine, but I believe my method also works.http://minds.wisconsin.edu/handle/1793/67009

3.14 If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by

$$\sigma_n = \dfrac{s_0+s_1+...+s_n}{n+1} (n=0,1,2,...).$$

(d) Put $a_n=s_n-s_{n-1}$, for $n\geq 1$. Show that

$$s_n-\sigma_n=\dfrac{1}{n+1}\sum_{k=1}^n k a_n$...$(1)$$

Assume that lim$(n a_n)=0$ and that ${\sigma_n}$ converges. Prove that $\{s_n\}$ converges.

Thanks for your suggestion. Although I didn't know this theorem before, surprisingly, I unconsciously proved a part of the theorem in the question (a). I'm really glad to know a simpler way to solve the question.
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user12345May 20 '14 at 1:07